| L(s) = 1 | + 2.13·2-s + 2.54·4-s + 1.76·5-s + 2.12·7-s + 1.15·8-s + 3.75·10-s + 0.197·11-s + 5.26·13-s + 4.52·14-s − 2.62·16-s + 4.06·17-s − 19-s + 4.47·20-s + 0.420·22-s + 2.60·23-s − 1.90·25-s + 11.2·26-s + 5.39·28-s + 2.04·29-s + 1.48·31-s − 7.89·32-s + 8.66·34-s + 3.73·35-s + 1.49·37-s − 2.13·38-s + 2.03·40-s + 4.45·41-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.27·4-s + 0.787·5-s + 0.801·7-s + 0.407·8-s + 1.18·10-s + 0.0595·11-s + 1.46·13-s + 1.20·14-s − 0.656·16-s + 0.985·17-s − 0.229·19-s + 1.00·20-s + 0.0896·22-s + 0.544·23-s − 0.380·25-s + 2.20·26-s + 1.01·28-s + 0.379·29-s + 0.267·31-s − 1.39·32-s + 1.48·34-s + 0.631·35-s + 0.246·37-s − 0.345·38-s + 0.320·40-s + 0.695·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.088940717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.088940717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 - 0.197T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 4.06T + 17T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 - 2.04T + 29T^{2} \) |
| 31 | \( 1 - 1.48T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 - 1.68T + 43T^{2} \) |
| 53 | \( 1 + 4.07T + 53T^{2} \) |
| 59 | \( 1 - 0.877T + 59T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 - 6.13T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 1.42T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 + 0.344T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.74783554267565797115222494869, −6.79559614019247940870502409827, −6.10003672072370493414112918545, −5.74511474661650290507979794934, −5.02684216014154032930166710942, −4.36935277769599646238057049156, −3.61314492639532193079601181332, −2.91221427910082859585462952362, −1.96137815214986557557434061559, −1.15432679494835921465481179588,
1.15432679494835921465481179588, 1.96137815214986557557434061559, 2.91221427910082859585462952362, 3.61314492639532193079601181332, 4.36935277769599646238057049156, 5.02684216014154032930166710942, 5.74511474661650290507979794934, 6.10003672072370493414112918545, 6.79559614019247940870502409827, 7.74783554267565797115222494869
